We compute the trace of an endomorphism in equivariant bivariant K-theory
for a compact group \(G\) in several ways: geometrically using geometric
correspondences, algebraically using localisation, and as a Hattori--Stallings
trace. This results in an equivariant version of the classical Lefschetz
fixed-point theorem, which applies to arbitrary equivariant correspondences,
not just maps. \textit{We dedicate this article to Tamaz Kandelaki, who
was a coauthor in an earlier version of this article, and passed away in
2012. We will remember him for his warm character and his perseverance
in doing mathematics in difficult circumstances.}